Coarse-grained models for macromolecular systems...Collection SFN 12 (2011) 285–300 C Owned by the authors, published by EDP Sciences, 2011 DOI: 10.1051/sfn/201112014 Coarse-grained

Collection SFN 12 (2011) 285–300C© Owned by the authors, published by EDP Sciences, 2011DOI: 10.1051/sfn/201112014

Coarse-grained models for macromolecular systems

V. Hugouvieux

INRA, UMR1083 SPO, 34060 Montpellier Cedex 1, France

Abstract. Neutron scattering experiments and simulations are often used as complementary tools in viewof revealing the structure and dynamics of molecular and macromolecular systems. For polymeric and self-assembling systems, the simulation of large-scale structures and long-time processes is often achieved byusing coarse-grained models which allow to gain some orders of magnitude in space and time scales. Bydiscarding some details of the chains, they also allow a better understanding of the main features of thesystem that govern its behaviour, such as the sequence of a macromolecule or some interaction that leadsto its self-assembly. After a brief introduction on coarse-grained models and associated representations, theapproach is illustrated with the case of amphiphilic regularly alternating multiblock copolymers in diluteand semidilute solutions. In this context, a generic HP (H: hydrophobic ; P: hydrophilic, polar) model isused together with a lattice representation of the system and a Monte Carlo algorithm. The simulations giveaccess to the various structures and phases of the system as a function of the energy of interaction betweenH monomers, the ratio of H monomers in the chain, the length of the blocks and the concentration. In dilutesolution structures range from swollen coils in good solvent to chains of micelles and layered structuresin poor solvent. In semidilute solution microphase separation and gelation are observed as a function ofsubstitution ratio and concentration.

1. WHY USE COARSE-GRAINED OR MESOSCOPIC MODELS?

One of the main reasons why scientists use coarse-grained or mesoscopic descriptions of molecularsystems is to achieve longer simulation times and larger spatial scales by discarding part of the chemicaldetails of the system. In the case of polymers and self-assembling systems this scale issue is fundamentalas the behaviour of macromolecules covers several orders of magnitude in space and time. Severalreview articles deal with the application of coarse-grained models to soft materials [1], proteins [2] andmembranes [3].

For standard condensed matter systems such as simple liquids it is often sufficient to performsimulations of some 103 atoms interacting through realistic forces. Even with such small systems thestructural features and correlations can be captured: as shown by the oscillations of their pair correlationfunction, these systems are usually homogeneous on scales of the order of 10 Å. Quantum chemistrysimulations are moreover available for these systems and enable relevant force fields on the atomic scaleto be derived.

On the polymer side [4], the situation is quite different with relevant length scales encompassingseveral orders of magnitude. At the level of a single chain of some 103 monomers, bond lengths are ofthe order of 1 Å, persistence length is some 10 Å and the radius of gyration of the coil is about 100 Å.For collective phenomena, cross-linked networks and phase separation typical length scales may evenbe larger and reach 103 Å. Also the size of the simulation box has to be larger than the characteristiclength scales of the system, which leads to very large simulated systems (106 atoms or more). Hence theatomic scale simulation of polymer systems is restricted to small values of the degree of polymerizationNP and systems far from any phase transitions.

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0,which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work isproperly cited.

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The need for coarse-grained representations is emphasized by the dispersion of the time scalespertaining to macromolecular systems. While the characteristic time for a C-C bond length or anglevibration is around 10−13 s and the average time between two reorientational jumps in the torsionalpotential is around 10−11 s, the time for the relaxation of the coil configuration of a single chain ismany orders of magnitude larger. This relaxation time �NP

scales with the chain length as �NP∝ N2

P forNP < 100 (Rouse model) [5] while for longer chains the scaling becomes �NP

∝ N3P (reptation model)

[5, 6]. This typically leads to relaxation times of the order of 10−8 to 10−5 s, to be compared with timesfor bond length vibrations or local reorientations.

To cope with the broad range of time and length scales, coarse-grained or mesoscopic representationsof polymer or self-assembling systems (such as membranes) have to be envisaged. Such models areobtained by lumping groups of atoms together into single effective particles or interaction sites, thusreducing the number of degrees of freedom of the system and removing the faster vibrations. Alsosimplified and softer potentials are used for the interactions between these particles (electrostatics andtorsions are often omitted). This leads to an appreciable speed-up of the simulations and gives access tocollective phenomena such as the self-assembly of block copolymers in solution [7, 8] or the formationand thermodynamic stability of membranes [3], usually not available from atomistic simulations.

Obviously coarse-grained models are not able to capture all the atomic scale properties of the systemunder study. This is precisely the point of using such a representation: it enables the main featuresinfluencing the mesoscale behaviour to be elucidated. Therefore two main kinds of questions can beaddressed through mesoscopic modelling and simulation. On the one hand insight can be gained intothe interactions that are responsible for the mesoscale behaviour (phase separation, self-assembly). Onthe other hand the universal behaviour at a mesoscopic scale can be revealed (e.g. phase transitions indiblock copolymer melts, transitions between different morphologies of copolymers in solution).

2. HOW TO DESIGN A MESOSCALE MODEL?

When building a mesoscale model, one of the main issues is to determine which degrees of freedom orinteractions of the system have to be retained in order to recover the behaviour of interest. Two maintypes of coarse-grained models can be distinguished : the minimal or generic ones, and those derived bysystematic coarse-graining. A generic polymer model is able to capture the features common to a classof polymers with a given architecture and composition pattern. In contrast, two polymers with similararchitectures but chemically different monomers should behave differently in a systematically coarse-grained representation.

Before describing coarse-grained models developed for macromolecules, let us first focus on the caseof the solvent, which accounts for a substantial part of the degrees of freedom of the system. In coarse-grained simulations of polymers, explicit description of the solvent is often eliminated and replacedby effective interactions between the solute molecules. We talk about solvent-free or implicit solventmodels. For instance, the radius R of a polymer chain consisting of N monomers scales like R ∝ N1/2

and the size L of the box should be of the order L ∼ R. So the mass fraction of polymer, N/L3 ∼ 1/L,decreases with increasing length of the polymer. Hence discarding the explicit representation of thesolvent enables substantial computer simulation time to be saved.

In the case of minimal models (also called generic or simplified models) only a very restrictedset of features of molecules or macromolecules are taken into account. Besides excluded volume andconnectivity constraints, some interactions are deemed essential for the behaviour of interest. Theseare retained in the model. This kind of models is widespread for the study of scaling properties ofpolymeric systems (both static and dynamic) and in the field of self-assembling amphiphiles. Amonggeneric models, a distinction can be made between off-lattice and lattice representations.

The idea behind lattice representations is to describe the chains as self-avoiding walks on a lattice [9].The simplest model consists of a regular cubic lattice where each effective particle of the chain sits on alattice site. A bond connecting two beads of the chain is a link between two nearest-neighbour sites of

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Figure 1. Some standard lattice moves: a) 180◦ crankshaft; b) 90◦ crankshaft (d ≥ 3); c) slithering snake; d) pivotmove (see [9] for other kinds of moves).

Figure 2. Schematic representation of the bond fluctuation model (left) and bead-spring model (right).

the lattice. Single occupation of the sites ensures excluded-volume constraints and non-crossing of thestrands also has to be checked for. A variety of moves exist which are designed for different purposes, forinstance end-bond, kink-jump, crankshaft, slithering snake and pivot moves (see Figure 1). The latteris more relevant when studying static properties, since the aim of pivot moves is to rapidly generatedynamically uncorrelated configurations of the chains.

The most widely used lattice algorithm is the bond-fluctuation model [10, 11] (see Figure 2, left).It can be seen as an intermediate between cubic lattice models and continuum models, since the bondbetween two monomers can take many different values (36 in 2d and 108 in 3d, compared to 4 in 2d and6 in 3d for a simple cubic lattice). The constraints of excluded-volume and non-crossing of the strandsare assured by restrictions in the allowed bond lengths.

In off-lattice representations, the positions of effective monomers are no more restricted to latticesites. Again several models exist : freely jointed chain with rigid bonds, pearl-necklace model [4]. Thebead-spring model is the most widely used off-lattice model. Bonds are represented using harmonicpotentials with a constraint of finite extensibility (FENE potential) and effective particles interactthrough the Lennard-Jones potential (see Figure 2, right).

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Such generic models have been widely used for the study of the static and dynamic properties ofpolymers. For instance, off-lattice models have been applied to study the phase diagrams of diblockcopolymers [12] and the crossover from Rouse to reptation dynamics in polymer melts [13, 14]. Usingbead-spring models, the formation of micelles has been investigated either in solution [15] or at surfaces[16]. Lattice models can also prove helpful in the study of the micellization and phase separationof surfactants and multiblock copolymers [17–19]. Protein folding and aggregation is another fieldof application of these models, where they are used in view of elucidating the generic mechanismsinvolved [20].

The other option is to derive a coarse-grained model for a specific atomic system [21]. The generalprinciple of this method, usually called systematic coarse-graining, consists of:1. choosing a set of key features of the system that the coarse-grained model should reproduce and2. determining the coarse-grained interactions between the effective beads in order to reproduce the key

characteristics of the system.Two main kinds of such key features are presently used : structural and thermodynamic ones. Whencoarse-grained potentials are built using structural properties from all-atom molecular simulations,distributions of geometric quantities are used to compare the structure of the coarse-grained systemand that of the all-atom configurations. These can be intramolecular (distances between two adjacenteffective particles, bond angles between three consecutive effective beads, radius of gyration of thechain and eigenvalues of the corresponding tensor) or intermolecular (distances between the centersof mass of different chains) properties. The determination of the coarse-grained potentials is oftenbased on Boltzmann inversion (for instance, in the case of non-bonded interaction potentials, UCG(r) =−kT ln g(r) and variants thereof) of the target properties. When the target of the coarse-grained modelare the thermodynamic properties, parameters of the coarse-grained potentials can be chosen so asto reproduce free energies of vaporization, hydration or partitioning, as was done in the case of theMARTINI coarse-grained lipid model [22]. A presently active field of research is to determine whetherit is possible to derive coarse-grained potentials that would be consistent with both structural andthermodynamic properties of the corresponding all-atom description.

Systematic coarse-graining can for instance be used in conjunction with the method of dissipativeparticle dynamics [23].

Starting from atomistic descriptions and reaching coarser scales is the basic idea of multiscalesimulations, where phenomena are studied at different levels of resolution. These multiscale descriptionsenable a mapping to be done, from the high-resolution to the low-resolution scale, in order to reach largetime and space scales. On the other hand, back-mapping can also be envisaged from the coarse-grainedto the more-detailed level [24].

In the next section a case study is presented in details. It deals with the behaviour of amphiphilicmultiblock copolymers in dilute and semidilute solutions. These systems are studied using a genericlattice model and a Metropolis Monte Carlo algorithm.

3. MULTIBLOCK COPOLYMERS IN SOLUTION: MODEL AND SIMULATION METHOD

In this section a more detailed account is given of the self-assembly and phase behaviour of linearmultiblock copolymers in solution. The macromolecules are represented on a lattice at a coarse-grainedlevel and a standard Monte Carlo algorithm is used for the simulations.

3.1 Motivation: the behaviour of methylcellulose

Methylcellulose is a water-soluble cellulose derivative which belongs to the class of associatingpolymers. It is obtained by the substitution of some hydroxyl groups of the anhydroglucose units ofcellulose by the methoxide group O-CH3 (see Figure 3). Depending on the degree of methylationof a glucose monomer, the monomer can be considered hydrophilic (no substitution) or hydrophobic

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Figure 3. A generic monomer unit of methylcellulose with Ri = H or Ri = CH3.

(3 substituted groups). Methylcellulose is characterized by its average substitution degree whichcorresponds to the average number of substituted groups per monomer. The substitution patterncan vary strongly depending on the conditions for cellulose modification and it has been shownthat commercial methylcellulose displays a heterogeneous distribution of the methyl groups alongthe cellulose backbone. In agreement with experimental results [25–27], methylcellulose is usuallyconsidered as a multiblock copolymer and not as a statistical one. The multiblock pattern gives anamphiphilic character to methylcellulose chains. Also this biopolymer derivative is known to performthermoreversible gelation upon heating [27]. The interplay between amphiphilicity and gelation abilitygives rise to a range of structures and properties which attract a lot of interest from food, biomedical,pharmaceutical and cosmetic industries. In particular amphiphilic block copolymers have attracted agreat deal of interest from the biomedical field [28], with applications such as drug delivery vectors[29], nanoparticle stabilizers, nanoreservoirs, emulsion stabilizers, wetting agents, rheology modifiers[30, 31] or as injectable scaffold materials for tissue engineering [32]. Using a generic coarse-grainedmodel and Monte Carlo simulations, we want to improve our understanding of the behaviour ofamphiphilic multiblock copolymers in solution.

3.2 Defining a model

In view of understanding the impact of the substitution pattern and ratio on the behaviour ofmethylcellulose, copolymers are modelled as chains of Nm connected monomers of two types, eitherhydrophobic (H) or hydrophilic (P), depending on the number of substituted hydroxyl groups. We ignorethe chemical details of the monomer units and each of them is represented as a single bead.

As stated before, experiments tend to show that methylcellulose has a blocky chemical structure,with highly substituted zones along the chain. This is why in our model copolymers are represented asregularly alternating blocks of hydrophobic and hydrophilic monomers. They are denoted as (HBH

PBP)n,

where BH is the number of H monomers per hydrophobic block, BP the number of P monomers perhydrophilic block and n the number of HBH

PBPpatterns in the polymer. A given multiblock copolymer

thus consists of a total of Nm = n(BH + BP ) monomers and its hydrophobic substitution rate, Psub, isdefined as:

Psub = BH

BH + BP

(1)

Both types of monomers are subjected to excluded-volume constraints. The solvent effect is simulatedimplicitly and the interactions between the monomers and the solvent are accounted for indirectly. Aneffective short-range attractive interaction between the H monomers mimics the effect of hydrophobicitywhich favours the association of the H monomers. In the model the interaction is restricted to the nearestH neighbours which corresponds to the range of the repulsion between the H monomers and the solvent.This so called hydrophobic interaction strongly depends on the quality of the solvent with respect to theH monomers. In practice this interaction is taken into account through the effective energy Ei betweenneighbouring H monomers: when Ei = 0, there is no interaction between H monomers and the quality

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of the solvent is equally good for both kinds of monomers; for increasing |Ei |, the solvent becomesincreasingly poor for H monomers which leads to the formation of H clusters. The notion of goodand poor solvent always refers to H monomers, P monomers are always in good solvent conditions. Insummary, the interaction is always equal to zero between two hydrophilic monomers and between ahydrophilic and a hydrophobic monomer while there is an attractive interaction of strength Ei betweentwo hydrophobic monomers. Hence our model takes into account different solvent qualities and varioussizes and ratios of hydrophobic and hydrophilic blocks.

3.3 Lattice model and Monte Carlo

The behaviour of the multiblock copolymers is simulated using the algorithm of coarse-grained cellpolymer dynamics developed by Kolb and Axelos [33].

The basic idea of this model is to replace the complicated continuum dynamics of a polymer solutionor melt by a truncated lattice version. Consider any continuum polymer system. To discretize this systemwe regularly divide space into compact cells. The centers of these cells form a regular lattice. Thepolymer conformation and its dynamics can then be uniquely discretized by placing all the monomersthat lie in a given cell onto the corresponding lattice site. The cell size is a free parameter, for the presentcalculation we set it equal to two monomer volumes, i.e. each lattice site can be occupied by zero, one ortwo monomers. After projection of the conformation onto the lattice, the continuum dynamics becomesa nearest neighbour hopping dynamics. The bond lengths of the original model are also discretized to thevalues zero or one lattice spacings. This corresponds to a variation of the effective average bond lengthbetween one half and one lattice spacings, for a chain of doubly or singly occupied sites respectively.

The original polymer model and the lattice model have the same properties on scales larger than alattice spacing. Computationally a lattice model is much more efficient, as the complicated continuumdynamics is replaced by a simple nearest neighbour hopping dynamics on the lattice. In order toreproduce statics and dynamics correctly, a lattice model must respect three essential features of polymerstructure and dynamics: the monomer connectivity along the chain, the excluded volume interaction andthe non-crossing of polymer strands. The connectivity requirement is assured by construction becauseneighbouring monomers along the chain always sit on the same or on nearest neighbour sites. Theexcluded volume restriction is respected automatically because each site can be occupied by at mosttwo monomers. The condition that two polymer strands are not allowed to cross would require elaboratetesting for the proposed discretization. To avoid this, we change the rules slightly: the double occupancyof a lattice site is restricted to monomers that are nearest neighbours along the same polymer chain(chemically bonded monomers). This is a minor change, which effectively corresponds to an increaseof the range of the excluded volume interaction, but which allows to guarantee the strict non-crossingof strands in a simple way.

The key difference between cell polymer dynamics and standard lattice models is that in our modela lattice site can be occupied by two monomers. This is also its main advantage: pure reptation alongthe polymer chain is explicitly possible even at the highest densities, monomers moving between adoubly occupied and an empty site naturally create local chain length fluctuations on the lattice, thereare no locked up conformations and hence no ergodicity problems as for standard lattice models. Themodel is computationally efficient as it has, under athermal conditions, no adjustable parameters andonly one move type. The unique space and time scales of this move correspond to the cell size and itscharacteristic relaxation time of the corresponding real polymer system. The model is designed in a waysuch that the cited three basic conditions of polymer dynamics are guaranteed by construction for everymove. It can therefore be implemented as easily and as efficiently as the simplest lattice models.

Figure 4 shows a two-dimensional (hexagonal) representation of the scheme. The choice of the fccor hexagonal lattice is motivated by a greater number of nearest neighbours (12 resp. 6) and a widerand smoother range of bonding angles than a standard cubic (6) or square (4) lattice. For a preciseoperational definition of cell polymer dynamics the reader is referred to Ref. [34].

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Figure 4. 2D representation of a polymer on the lattice and the possible monomer moves. Black dots and boldlines represent an allowed conformation of the chain and its monomers. Green lines and dots are acceptable movesof end monomer (e1, e2) resp. internal monomers (i1, i2, i3). Red lines and dots (x1, x2) show forbidden moves(From [34]).

In this algorithm, Rouse type moves dominate the dynamics for dilute solutions whereas reptationmoves dominate the dynamics at melt densities, just as expected for real polymers [5]. The shift fromRouse dominated to reptation dominated dynamics occurs naturally upon increasing the monomerdensity and without the tuning of a parameter.

Examples of moves that are rejected because they would lead to forbidden conformations are alsoshown in Figure 4 (internal monomer x1, end monomer x2). Interactions are introduced by assigningevery monomer to be H or P and by adding an energy contribution Ei < 0 to the total energy for everypair of H monomers that either occupy the same site or that occupy two nearest neighbour sites. Whentrying to move an interacting hydrophobic (H) monomer, the move is only accepted if it is energeticallyfavourable, according to the Metropolis sampling scheme. The (dimensionless) energy difference �E

of a conformation before and after a move is calculated and the move is accepted if all other constraintsare satisfied and if min(1, exp(−�E)) < ran, where 0 ≤ ran ≤ 1 is a uniformly distributed randomnumber. Time is measured in Monte Carlo steps (MCS) which corresponds to NmNp trial moves, whereNm is the total number of monomers per chain and Np the number of chains in the system.

4. DILUTE SOLUTION OF MULTIBLOCK COPOLYMERS

In the case of dilute solutions of copolymers, no intermolecular interactions are included and thereforewe may consider the properties of a single copolymer. Simulations give access to different kinds ofinformation. Direct visual inspection of the conformations and their evolution with time gives a firstinsight into the structure and dynamics of the mesostructures as a function of the parameters. A morequantitative analysis can be made from calculating structural properties of the simulated configurations:radii of gyration, size of H clusters and form factors. Both the radii of gyration and the form factorsare also available from experiments. Confronting visual inspection and statistical properties allowsone to identify the signature of each conformation that is expected in an experimental measurement(e.g. scattering experiment).

4.1 Effect of solvent quality

We first focus on the effect of the interaction energy Ei between H monomers on the overall size of thechain. The copolymer is characterized by its radius of gyration Rg , computed as:

R2g = 1

Nm

⟨Nm∑i=1

(�ri − �rcm)2

⟩(2)

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0.5 1 1.5E

i

0

50

100

150

200

250

Rg2

0.5 0.6 0.7 0.8 0.9E

i

0.8

0.85

0.9

0.95

1

αFigure 5. Mean squared radii of gyration as a function of Ei , for Nm = 600 and BH = 3. Dotted line: Psub = 0.1;Dashed line: Psub = 0.3; Full line: Psub = 0.5. Inset: Ratio � = R2

gH /R2gP versus Ei for Psub = 0.3 (From [34]).

where �ri is the position of monomer i and �rcm is the position of the center of mass of the polymer ata given time, and 〈.〉 denotes an average over all configurations. Figure 5 shows the evolution of theaverage value of the squared radius of gyration of the copolymer with the interaction energy Ei , forchain length (Nm = 600) and H block size (BH = 3), for three values of Psub (Psub = 0.1, 0.3 and 0.5).We observe the expected behaviour, from a swollen state with large R2

g at low values of Ei (goodsolvent) to a more compact state with high solvent selectivity (poor solvent for H monomers).Decreasing the length of the non-interacting P blocks while keeping the H block length fixed (i.e.increasing Psub) shifts the collapsed region to weaker energies. The same effect is observed whenincreasing simultaneously the lengths of interacting and non-interacting blocks while keeping Psub fixed(not shown here), in agreement with previous findings [35].

Also form factors can be computed using the simulated configurations :

F (q) = 1

Nm

⟨Nm∑

j ,k=1

exp [i�q · (�rj − �rk)]

⟩(3)

Since the system is isotropic, the form factor can be averaged over all �q vectors of equal length, usingthe scheme described in Ref. [36]. The average 〈.〉 is also performed over all configurations. We limitthe analysis of the form factor at large q to q < 2�/a, where a is the lattice spacing, as for q � 2�/a

the scattering intensity is dominated by lattice effects. We assume here that the scattering lengths ofboth kinds of monomers are equal. Figure 6 shows the form factors of the chains under different solventconditions and for two values of Psub. At Ei = 0, the solvent is equally good for H and P monomers andthe copolymer shows a fractal dimension of 1.7 in the intermediate q-range, typical of homopolymersin good solvent with F (q) ∝ q−1.7. For interaction energies typical of poor solvents (as determinedfrom the energy dependence of Rg , see Figure 5), several features can be noticed. The low q region ofF (q), for qRg 1, gives access to the overall size of the copolymer, using the Guinier approximation.Figure 6 is in agreement with the observations of the evolution of Rg , with a compaction of the coil asEi increases, for fixed Psub: In a poor solvent the plateau is reached at larger q values (smaller valuesof Rg). In the intermediate q-range, for qRg ≥ 1, one probes the internal structure of the polymer. Inparticular, F (q) ∝ q−4 characterizes a compact spherical structure. A homopolymer (Psub = 1) underpoor solvent conditions is expected to collapse into such a compact spherical structure (see Figure 6).For Psub values different from Psub = 1 (red line in Figure 6), the behaviour is different and correlationsappear at intermediate q values, which become more pronounced with increasing Ei at fixed Psub (notshown).

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0.1 1q

0.1

1

10

100

1000

F(q)

q-1.7

q-4

Figure 6. Total form factors for different values of Psub with Nm = 600 and BH = 3. Good solvent conditions:Ei = 0.0 (black line). Poor solvent conditions: Psub = 0.3, Ei = 0.9 (red line); Psub = 1, Ei = 0.4 (blue line)(From [34]).

0 10 20 30Cluster size

0

0.1

0.2

Prob

abili

ty

Ei = 0.5

Ei = 0.6

Ei = 0.7

Ei = 0.8 0.5 0.6 0.7 0.8

Ei

0

0.5

1

Figure 7. Probability distribution (number probability) of cluster sizes and their evolution with interaction energy,from Ei = 0.5 (good solvent) to Ei = 0.8 (poor solvent). The copolymer is defined as (HBH

PBP)60 with Nm = 600,

BH = 3, BP = 7 and Psub = 0.3. The cluster size is expressed in numbers of hydrophobic blocks. The probabilityis negligible beyond Nc ≥ 30. The inset shows the total fraction of small clusters (Nc < 5) as a function of Ei

(From [34]).

The presence of correlation lengths in F (q), as shown by the shoulder at q ≈ 1, suggests theexistence of mesostructures with characteristic lengths smaller than the overall chain size Rg in poorsolvent. This observation is likely to be related to the existence of dense clusters of H monomers, asindicated by the decreasing ratios of hydrophobic to hydrophilic radii of gyration (Figure 5). To checkthis hypothesis we have identified hydrophobic clusters in the simulated conformations. For our purposea cluster is defined as a set of interacting H blocks. Two blocks are said to interact if any two monomersof these blocks are nearest neighbours. This property is transitive, if block A interacts with block B andblock B with block C then all three belong to the same cluster. The aggregation number NC of a clusteris defined as the number of H blocks belonging to this cluster.

From the cluster size distributions for different values of Psub and Ei , and for sufficiently longchains (Nm = 600), we distinguish two main types of behaviour. For low Psub, in a poor solvent adistribution of cluster sizes appears around a well defined average value Nc. With increasing energy Ei

the peak of the distribution becomes sharper and shifts to higher values. This is particularly clear forPsub = 0.3 (see Figure 7). For Ei = 0.8 there are 〈Nc〉 ≈ 20 blocks in a cluster which corresponds to an

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Figure 8. Typical configurations for different substitution rates: (a) Psub = 0.3, chain of micelles; (b) Psub = 0.5with BH = 3, tubular structure; (c) Psub = 0.5 with BH = 2, layered structure. H monomers are depicted in black,P monomers in light gray (From [34]).

average of three H clusters per chain. For Psub = 0.1 the same general picture is observed, but with theinteraction energies shifted to higher values. Simulations with Nm = 600 and 1200 at Psub = 0.1 supportthe expectation that the number of clusters is growing with the chain length and the size distribution(aggregation number) is independent of chain length. For large values of Psub, Psub = 0.5, the clustersize distribution is qualitatively different from small Psub. Up to Ei = 0.4 no well defined cluster ispresent, beyond Ei = 0.5 almost all H blocks collapse into a single cluster. Not surprisingly, in bothcases a decrease of the probability of finding isolated H blocks with increasing interaction energy Ei isobserved.

Calculation of the radii of gyration, form factors and cluster size distributions gives informationabout the influence of the solvent quality on the overall structure of the polymer coil but also on itsinternal correlations. For a full understanding of the intramolecular structures in poor solvent, furtheranalyses have to be performed.

4.2 Intramolecular structures in poor solvent

A first insight into the intramolecular structures in poor solvent can be obtained from the directvisualization of copolymer conformations. We notice several well-defined features, as shown in Figure 8.

For Psub ≤ 0.3, we observe the formation of intramolecular chains of micelles (a singleintramolecular micelle for short chains): The H blocks aggregate into clusters (see Figure 7) whichare surrounded by P blocks (see Figure 8a). These clusters typically are spherical or ellipsoidal, andthey are linked to each other by one or several P blocks. The number of such intramolecular clustersincreases linearly with Nm.

For Psub ≥ 0.5, when decreasing the solvent quality, the copolymer undergoes a transition from aswollen chain to a chain containing H aggregates and then to a single cluster containing all H blocks,no matter how long the copolymer. There are two different types of structures observed in the very poorsolvent single cluster regime with Psub = 0.5. For BH = BP = 3, upon increasing the energy there firstappears a tubular shaped single cluster (the tube may be open, torus-like or with its ends folded backanywhere onto the tube) and for stronger energies a layered structure with a hydrophobic inner layer(crystalline, possibly distorted and with defects), see Figure 8. The transition from tubular to layeredstructure with increasing Ei proceeds via the formation of a chain of growing pieces of the layeredstructure. The transition from the tubular to the layered structure is most clearly seen in simulations

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0.1 1q

0.01

1

100

F H(q

)

q-2

(a)

(b)

(c)

(d)

q-4

Figure 9. Partial form factor of hydrophobic monomers under poor solvent conditions. From bottom to top:(a) Psub = 0.3, Nm = 600, BH = 3, BP = 7, Ei = 0.9 (chain of micelles); (b) Psub = 0.5, Nm = 600, BH = 3,BP = 3, Ei = 0.7 (tubular structure); (c) Psub = 0.5, Nm = 600, BH = 2, BP = 2, Ei = 0.7 (layered structure);(d) Psub = 0.625, Nm = 600, BH = 5, BP = 3, Ei = 0.8 (layered structure with rough surfaces and defects). Solidline: analytical form factor of a 2d-platelet. Note that the curves are normalized by NH and shifted vertically by oneorder of magnitude for clarity (From [34]).

using a slowly increasing energy ramp. The formation of a tubular and then a layered structure uponincreasing the energy is also observed for BH = 5 and BP = 3. On the other hand, for BH = BP = 2,there is no tubular structure and the layered cluster is formed directly with poorer solvent quality. In thiscase the layered structure is much more regular and the surfaces are much smoother than the previousones.

A more detailed analysis of the intramolecular structures can be obtained from the partial structurefactors FH (q) (see Figure 9), Fcluster (q) (not shown) and FP (q) (see Figure 10). FH (q) (FP (q)) givesinformation about the correlations between all H (all P) monomers, whereas Fcluster (q) correlates Hmonomers within the same cluster and can be used for characterizing the shapes and sizes of the Hclusters (see Sec. 4.1). Let us first focus on the properties of FH (q) (Figure 9) and Fcluster (q). Forsmall q values, FH (q) reaches a plateau, which corresponds to the Guinier regime where qRg 1. Forintermediate q values, FH (q) tends to behave as q−4, which stands for a compact object with a sharpinterface (well defined density drop). This feature is independent of Psub and in this range of q values,FH (q) and Fcluster (q) can be superimposed (adjusting the scales appropriately). Note that for Psub > 0.3,FH (q) ≈ Fcluster (q) as most H blocks belong to a single cluster. For Psub = 0.3, Fcluster (q) can be fittedto the analytical form factor of a polydisperse sphere of radius Rs = 1.85 (not shown, see [34]). Forlower q values, the behaviours of FH (q) and Fcluster (q) depart from each other for Psub = 0.3: FH (q)shows a well-defined correlation distance at qc ≈ 1.2 (≈ 5.2 lattice units in real space), which becomesmore pronounced for longer chains and disappears for short chains (single intramolecular micelle).Therefore this correlation peak reflects the average distance between H clusters: It is absent in the caseof a single cluster and increases when the number of clusters and hence the number of inter-clusterdistances increases. For Psub = 0.5 and BH = 3 (see Figure 9b), we observe an inflexion point aroundq ≈ 1.3, which turns into a short plateau for longer chains (Nm = 1200, not shown here). As explainedabove (and detailed in Sec. 4.1), there is a single cluster of H monomers and this inflexion point mightcorrespond to the mean distance between monomers on opposite sides of closed tubular conformations,predominant in this case. This hypothesis is supported by the fact that the intensity at which the inflexionoccurs increases with increasing Ei , when the structure becomes better defined. For Psub = 0.5 and

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0.1 1q

0.01

1

100

FP(q

)(a)

(b)

(c)

(d)

Figure 10. Partial form factors of hydrophilic monomers in poor solvent conditions. From bottom to top:(a) Psub = 0.3, Nm = 600, BH = 3, BP = 7, Ei = 0.9 (chain of micelles); (b) Psub = 0.5, Nm = 600, BH = 3,BP = 3, Ei = 0.7 (tubular shape); (c) Psub = 0.5, Nm = 600, BH = 2, BP = 2, Ei = 0.7 (layered structure); (d)Psub = 0.625, Nm = 600, BH = 5, BP = 3, Ei = 0.8 (layered structure with rough surfaces). Note that curves arenormalized by NH and shifted vertically for clarity (From [34]).

BH = 2, FH (q) becomes quite different with the onset of a q−2 behaviour at intermediate q values,which is typical of two-dimensional structures. For higher q, the structure is compact with a q−4 slope.The H blocks form a compact two-dimensional structure. The same kind of behaviour is observed forPsub = 0.625 with BH = 5 and BP = 3, however the onset of the q−2 behaviour is not observed for tworeasons: For the same value of Nm, the layer is thicker and its surface area is smaller than for BH = 2,which reduces the range of the q−2 behaviour. At the same time, the surface is much rougher and showsmore defects for Psub = 0.625, which leads to an effective surface fractal dimension dg ≥ 2.

If we now move on to the contribution from the P monomers (Figure 10), the salient feature of FP (q)is a peak at a qP value between 1 and 2, depending on Psub. The most straightforward case to interpretis the two-dimensional platelet of H blocks, as found for Psub = 0.5 with BH = 2 and Psub = 0.625with BH = 5. Due to connectivity constraints, the P blocks must coat the surfaces of the H plateletand consequently, FP (q) has a scattering function similar to that of stacked lamellae. The position andwidths of the peak is in this case related to the spacing d between two consecutive lamellae [37]. Forthe other two cases, the situation is quite different. The distances dp = 2�

qPcan be associated with the

tube diameter and the sphere diameter of the hole due to the presence of the H core (see [34] for moredetails).

4.3 Overview of the behaviour in dilute solution

Comparing the statistical properties extracted from the simulations with the direct visualization of theintramolecular structures, we are able to give an accurate description of the conformations of singlemultiblock copolymers as a function of solvent quality in terms of the form factors.

For weak interaction energies, the copolymer trivially has the properties of a swollen coil, asshown by F (q) ∝ q−1.7. The transition from good to poor solvent is accompanied by the formationof fluctuating aggregates due to the attractive interaction between H monomers. With increasing valuesof Ei , after the individual collapse of the hydrophobic blocks different intramolecular structures develop.This is shown in the phase diagram, Figure 11, as a function of energy Ei and hydrophilic block length

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Figure 11. Phase diagram of the intramolecular structures of multiblock copolymers as a function of Ei and 1BP

, forBH = 3. Symbols denote simulated systems, with equilibration at constant energy (solid symbols) or with an energyramp (open symbol). Copolymers in good solvent are denoted as triangles, pearl necklaces of micelles as dots,tubular structures as squares and layered structures as diamonds. Dashed lines indicate the boundaries between thephases corresponding to the different structures. Snapshots show typical conformations observed in the simulations.The arrow points in the direction where a simultaneous increase of Ei and BP maintains the energy-entropy balancenecessary for stabilizing a given structure (From [34]).

BP . With increasingly poorer solvent the qualitatively different conformations encountered are, in order,chains of micelles (see region A in Figure 11), tubular (region B) and layered structures (region C). Thediagram corresponds to a given value of BH (BH = 3 in Figure 11), but it is similar for other values ofBH . The parameters Ei and BP (or 1

BPin the diagram) are used, because the observed structures result

from a balance between energy and entropy: Ei governs the energy contribution of the interacting Hmonomers, while BP is a measure for the configurational entropy of the P blocks. Different values ofBH change the size of the elementary H cores, but do not change the principle of the energy-entropybalance responsible for the different structures.

In region A of Figure 11, intramolecular pearl necklaces of core-shell micelles are formed, withwell-defined core size of the micelles. The size of the pearls are stabilized by the P shells: Increasingthe size of the H clusters, for a given Ei and BP , would lead to an important entropy penalty due tothe crowding of the P shell; decreasing the size of the H cores would induce an energy loss that is notcompensated by a sufficient entropy gain. As a consequence, a simultaneous increase of Ei and BP

in region A (see the arrow) keeps the topology of the structure unchanged: strengthening Ei tends toincrease the size of the H cores, increasing the length of the P blocks reinforces the protective corona ofthe micelles and tends to decrease the core size.

In region B, the formation of tubular-shaped structures with a hydrophobic core is observed. Sucha tubular core is energetically more favourable than the spherical cores of region A but it has lesssurface area available for the hydrophobic corona. With respect to region A, region B corresponds eitherto copolymers with shorter P blocks (larger 1

BP) thus reducing the entropic contribution, or to larger

energies Ei , or both. As in region A, the length of the copolymer has no influence on the structure of thetube. The energy Ei does have an effect on the detailed shape of the tube: for smaller values the tubeand the corona have a spherical cross-section whereas for larger energies small parts of the tube locallybegin to form pieces of layered domains, as a precursor of the third morphology.

In region C one observes two-dimensional disk-like layered structures with an inner H layer and twoouter P layers. With respect to B this morphology requires a further increase of the energy Ei and/ora decrease of the length of the hydrophilic blocks BH . Besides the data point for BH = 3 shown inFigure 11 similar structures were obtained for Ei ≈ 0.7 for BH = BP = 2 and for BH = 5,BP = 3. The

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most perfect crystalline structures were obtained for BH = BP = 2 whereas for BH = 5 and BP = 3the resulting crystal is less perfect and with a rougher surface. This difference can be explained by thevery short H and P blocks for BH = BP = 2, which impose a strong restriction on the hydrophobicmonomers during the formation of the layer.

Simulations with other values of BH confirm the morphologies and their dependence on theparameters. The chain of transitions of aggregates → tubular shape → disk-like layered structure withincreasing Ei is not only observed for BH = BP = 3 but also for BH = 3 and BP = 5, with energiesEi = 0.5 and Ei = 0.7, respectively. The different morphologies and their order in the phase diagramcan be understood as a result of the competition between entropy and energy: starting from the origin,the same morphological changes are expected from an entropy decrease (horizontally) and from anenergy increase (vertically).

The different morphologies found in our simulations can be compared with the theoretical workbased on scaling arguments by Borisov and Zhulina [38] on amphiphilic graft copolymers withhydrophobic backbone: They predict the formation of analogous structures (necklace of star-like orcrew-cut micelles, cylindrical worm-like micelle and lamellar structure) depending on the degree ofbranching of the copolymer and the solvent quality. Despite the different connectivity constraints for thetwo polymers, sparse grafting may be compared to high substitution rate and dense grafting correspondsto multiblock copolymers with few associating monomers. On the experimental side, different studies onmultiblock linear copolymers and graft copolymers revealed the formation of intramolecular core-shellmicelles, chains of micelles and rod-like micelles (see [34] for more details), in qualitative agreementwith our findings.

5. SEMIDILUTE SOLUTION OF MULTIBLOCK COPOLYMERS

In semidilute solution both intra- and intermolecular interactions are accounted for. The concentration �is defined as the occupied fraction of the total available volume which, because of the double occupancy,is twice the number of available lattice sites. Simulations were performed on monodisperse many-chainsystems, for two substitution ratios, Psub = 0.2 and Psub = 0.5, for different concentrations � rangingfrom 0.01 to 0.5 and for different interaction energies Ei . This enables a broad range of behaviours to beencompassed, from rather dilute and good solvent conditions, to concentrated and strongly interactingsystems. Also different chain lengths Nm and numbers Np are used as they give insight into theirinfluence on phase behaviour (influence of Nm on percolation threshold) and finite-size effects (influenceof the box size at a given concentration). For poorly substituted copolymers micelles are formed, whichmay be either intra- or intermolecular depending on concentration. For the higher substitution ratio(Psub = 0.5), larger tubular structures form which grow with concentration. Also gelation is observed.The structure of the observed network strongly depends on Psub: for poorly substituted copolymers, theconnection is through the cross-linked micelles while for the highly substituted chains, the connectionis through the extended hydrophobic cores. The interplay between gelation and phase separation of thehydrophobic monomers is observed in the case of poorly substituted copolymers. More details aboutthis work can be found in [39].

Acknowledgements

The author would like to thank Drs. Max Kolb and Monique Axelos for fruitful collaboration while working on thebehaviour of multiblock copolymers.

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